Verifying Linear Independence

Linear dependence

We defined linear dependence as a collection with linearly dependent vectors if, given a collection of vectors $\bold x_1, \bold x_2, \cdots, \bold x_n$, there exists a combination of scalars $w_1,w_2,\cdots,w_n$ with at least one nonzero value such that

$$w_1\bold{x_1}+w_2\bold{x_2}+...+w_n\bold{x_n}=\bold{0}$$

There could be infinitely many possible linear combinations, and it’s not possible to check them one by one. We’ve already discussed a few special cases and tricks to check for linear dependence in a combination. This lesson will use elimination as a formal method to solve the vector equation and find the possible combinations.

Procedure

Our goal is to find the combination of scalars $w_1,w_2,\cdots,w_n$, such that

$$w_1\bold{x_1}+w_2\bold{x_2}+...+w_n\bold{x_n}=\bold{0}$$

Expanding the vectors, we get

$$w_1 \begin{bmatrix} x_{11} \\ x_{12} \\ \vdots \\ x_{1m} \end{bmatrix} + w_2 \begin{bmatrix} x_{21} \\ x_{22} \\ \vdots \\ x_{2m} \end{bmatrix} + \cdots + w_n \begin{bmatrix} x_{n1} \\ x_{n2} \\ \vdots \\ x_{1m} \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ \vdots \\ 0 \end{bmatrix}$$ ...